We consider a random \(\Delta \) -regular graph generated according to the configuration model [1]}, [2]}.
Given a partition of a set of \(\Delta n\)  points into \(n\)  cells \(v_1, v_2, \dots , v_n\)  of \(\Delta \)  points, a configuration is a perfect matching of the points into \(\Delta n/2\)  pairs. It corresponds to a (not necessarily connected) multigraph \(G^{\prime }\)  in which the cells are regarded as vertices and the pairs as edges: a pair of points \(\lbrace x, y\rbrace \)  in the configuration corresponds to an edge \((v_i, v_j)\)  of \(G^{\prime }\)  where \(x\in v_i\)  and \(y\in v_j\) .
Since each \(\Delta \) -regular graph has exactly \((\Delta !)^n\)  corresponding configurations, a \(\Delta \) -regular graph can be generated uniformly at random by rejection sampling: choose a configuration uniformly at random,To generate a random configuration, the points in a pair can be chosen sequentially: the first point can be selected using any rule, as long as the second point in that pair is chosen uniformly from the remaining points. and reject the result if the corresponding multigraph \(G^{\prime }\)  is not simple or not connected.
The configuration model enables us to show properties of a random \(\Delta \) -regular graph by analyzing a multigraph \(G^{\prime }\)  corresponding to a random configuration.
